Fourier Transform of Hermite polynomial $H_n(x)e^{-\frac{x^2}{2}}$

I'm trying to solve the 2D paraxial equation $$2i\partial_zu=-\partial_x^2u$$, for the initial condition $$u(x,z=0)=H_n(x)e^{-x^2/2}$$, with $$x$$ and $$z$$ both real and $$n\geq0$$.

For $$n=0$$, I used the Fourier transform—defined as $$\mathscr{F}_x\big\{f(x)\big\}(k)=\int_{-\infty}^{+\infty}dxf(x)e^{-ikx}$$ — to get

$$\tilde{u}(k,z)=\sqrt{2\pi}e^{-\frac{k^2}{2}(1+iz)}$$ $${u}(x,z)=\dfrac{1}{\sqrt{1+iz}}e^{-\frac{x^2}{2(1+iz)}}$$ in $$k$$ and $$x$$ domains respectively, i.e., I correctly get the propagation of a normalized Gaussian beam.

For the general case of $$n\geq1$$

1. How can I prove that $$\mathscr{F}\big\{H_n(x)e^{-x^2/2}\big\}=\sqrt{2\pi}(-i)^nH_n(k)e^{-k^2/2}$$?

2. What would be the inverse Fourier transform $$\mathscr{F}^{-1}\big\{\sqrt{2\pi}(-i)^nH_n(k)e^{-\frac{k^2}{2}(1+iz)}\big\}$$.

2 Answers

We can prove that:

$$I(k,a)=\int_{-\infty}^{\infty}H_n(ax)e^{-\frac{x^2}{2}}e^{-ikx}dx=\sqrt{2\pi}(-i)^n\Big(2a^2-1\Big)^{n/2}H_n{\Big(\frac{ak}{\sqrt{2a^2-1}}\Big)}e^{-\frac{k^2}{2}}$$

This shows that when $$a=1$$ we recover the desired result, that the Hermite polynomials are eigenfunctions of the Fourier transform. This also allows to compute the second Fourier transform in question since:

$$\int_{-\infty}^{\infty}H_n(k)e^{-\frac{k^2}{2b^2}}e^{ikx}\frac{dk}{2\pi}=\int_{-\infty}^{\infty}H_n(x)e^{-\frac{x^2}{2b^2}}e^{ikx}\frac{dx}{2\pi}=\frac{b}{2\pi}\int_{-\infty}^{\infty}H_n(b x)e^{-\frac{x^2}{2}}e^{ikb x}dx=(-1)^n\frac{b}{2\pi}I(kb,b)$$

and therefore we obtain the slightly more general result, valid for complex $$b$$ in general for which the integral converges:

$$I(x,a,b)=\int_{-\infty}^{\infty}H_n(ak)e^{-\frac{k^2}{2b^2}}e^{ikx}dk=i^nb\sqrt{2\pi}(2a^2b^2-1)^{n/2}H_n\Big(\frac{a~b^2 ~x}{\sqrt{2a^2b^2-1}}\Big)e^{-b^2x^2/2}$$

from which we finally obtain that:

$$\mathcal{F^{-1}}(\sqrt{2\pi}(-i)^nH_n(k)e^{-(1+iz)k^2/2})=\frac{(-1)^n}{\sqrt{1+iz}}\Big(\frac{1-iz}{1+iz}\Big)^{n/2}H_n\Big(\frac{x}{\sqrt{1+z^2}}\Big)e^{-\frac{x^2}{2(1+iz)}}$$

Note for calculation of $$I$$ to be added soon.

$$\textbf{EDIT:} ~~\small\text{Calculation of I(k,a,b) valid for all b\in\mathbb{C}}$$

First, write $$H_n(x)=\sum_{l}c_{nl}x^l$$. Substitute this in and perform the integrals:

$$I(k,a,b)=\sum_{l}c_{nl}a^l\int_{-\infty}^{\infty}x^le^{-x^2/2b^2}e^{-ikx}dx\\=b\sqrt{2\pi}\sum_{l}c_{nl}a^l\Big(i\frac{d}{dk}\Big)^le^{-k^2b^2/2}\\=b\sqrt{2\pi}\sum_{l}c_{nl}(iab)^le^{-k^2b^2/2}\Big[e^{k^2b^2/2}\Big(\frac{d}{d(kb)}\Big)^le^{-k^2b^2/2}\Big]\\=b\sqrt{2\pi}\sum_{l}c_{nl}(-iab)^le^{-k^2b^2/2}He_n(kb)$$

where $$He_n(x)$$ are the probabilists Hermite's polynomials as defined on the wikipedia page. Now utilize the represenations in terms of differential operators$$He_n(ax)=a^ne^{-D^2/2a^2}x^n, H_n(ax)=(2a)^ne^{-D^2/4a^2}x^n, D\equiv\frac{d}{dx}$$ repeatedly to rewrite again as:

\begin{align}I(k,a,b)&=b\sqrt{2\pi}e^{-k^2b^2/2}e^{-D^2/2b^2}\sum_{l}c_{nl}(-iab^2k)^l\\&=b\sqrt{2\pi}e^{-k^2b^2/2}e^{-D^2/2b^2}H_n(-iab^2k)\\&=b\sqrt{2\pi}e^{-k^2b^2/2}e^{-D^2/2b^2}(2iab^2)^n e^{D^2/4a^2b^4}k^n\\&=(2iab^2)^n e^{-k^2b^2/2}b\sqrt{2\pi}e^{-k^2b^2/2}e^{-D^2(1/2b^2-1/4a^2b^4)}k^n\\&=b\sqrt{2\pi}e^{-k^2b^2/2}\Big(-iab\sqrt{2-\frac{1}{a^2b^2}}\Big)^nH_n\Big(\frac{kb}{\sqrt{2-\frac{1}{a^2b^2}}}\Big)\end{align}

which upon some basic algebra yields the quoted result:

$$I(k,a,b)=b\sqrt{2\pi}(-i)^n\Big(2a^2b^2-1\Big)^{n/2}H_n\Big(\frac{kab^2}{\sqrt{2a^2b^2-1}}\Big)e^{-k^2b^2/2}$$

• Thanks for catching the typo, corrected now. Also I added the proof for the integral, which I found to be quite non-elementary. It would be nice to see someone with a less clunky proof. Commented Mar 24, 2020 at 0:24
• Thank you for your answer! I have been scratching my head for two days on this! Commented Mar 24, 2020 at 0:29
• The formula doesn't make sense for $a=1/\sqrt{2}$ [which occurs if you want to convert between probabilists' and physicists' Hermite polynomials]
– mkk
Commented Oct 2, 2022 at 13:36
• The limit $a\to 1/\sqrt{2}$ of the general formula makes perfect sense and recovers the correct formula $\propto k^n \exp(-k^2/2)$ when taken. Commented Oct 3, 2022 at 17:18

Perhaps this is not what you wanted, but it is clearest to me: some details about the quantum harmonic oscillator include that all eigenvectors are generated from $$u_1(x)=e^{-x^2/2}$$ by the "raising operator" $$R=i{\partial\over \partial x}ix$$, and that the $$n+1$$ eigenspace is exactly scalar multiples of $$R^n u_1$$. Since $$u_1$$ is annihilated by the lowering operator $$L=i{\partial\over \partial x}+ix$$, the interaction of Fourier transform with such operators shows that $$\widehat{u_1}=u_1$$ (at least with suitable normalization of Fourier transform.

Then an induction on $$n$$ shows that Fourier transform on $$R^n u_1$$ is multiplication by $$i^{-n}$$.

• Thank you Paul for your insight. I knew this was also Schrödinger's equation but haven't thought using the ladder operators. I'll keep the inductive solution in my back pocket! Commented Mar 23, 2020 at 23:35